This question is currently too broad. Could you please edit it and add more specifications? Like what is the expected input and output? Should the assembly be exported from CAD, or constructed using images? Please read this FAQ on what is required for a question to contain enough information?
– TymricMar 23 '15 at 18:23

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Those are good additions to your question. What operating system? Any other requirements or features? Any software you've already tried? Thanks!
– RockPaperLizard♦Mar 24 '15 at 0:11

Have you tried Catia V5 or Solid Works (or are these softwares too complex for your needs)? Also, do you consider in manualy do the assemble in your current software and capture your screen with another software and then edit the result?
– Fabio SilvaMar 24 '15 at 17:26

3 Answers
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I won't call this one the perfect answer, other than that it's free and available on multiple platforms. Consider Blender as an option for animating just about anything. It's been used to create short and full length "cartoon-type" animations, but certainly would be able to manage assemblies such as your animation.

If you have the models created elsewhere, it's simple enough to import them to Blender. At this point, my solid understanding goes out the window, but the general understanding kicks in.

Everything you need from lighting to keyframes to work flow is part of Blender. My experience with Blender has been for 3D modeling, and was able to manage the simple stuff. Plenty of complexity and a high learning curve, in my opinion, but FREE is a good start.

Cheetah3D should be able to do the trick. It has keyframe animation and flythrough animation through splines. All you need to do is group the to-be-assembled parts of your model and create paths how each part can be attached to its final destination.

If you want to stay on SketchUp, you could try an animation extension such as Keyframe Animation. You would have to setup the whole assembly manually exploded, afterwards frame each step, and finally export the animation from the file menu.

If you want something better suited for the particular task, consider Solid Edge.

Solid Edge is capable of creating such animations by combining the Explode, Animation and Render tools.

Solid Edge is a CAD software that focuses in the design of machines and mechanical parts. You would need to export the models in a format that Solid Edge supports (there are extensions to do the export - which I'm not mentioning because I can't test them right now - you may also use Blender to import SketchUp and export for Solid Edge).

Alternatively, you can recreate them. After all, SketchUp has already taught you the basics of CAD, I assure you will find Solid Edge easier than more traditional CAD solutions such as AutoCAD. This is in part to its "Synchronous Technology", which I would argue influenced SketchUp.

Using Solid Edge, what you do is as follows:

Auto explode the assembly.

Move the pieces to where you want to them to start, and out of the way of other pieces.

Open the animation tools (which automatically takes adds the previous steps you did) and adjust the timings for the animation, and add any other effects you want.

Render the animation to video.

Disclaimer: It has been more than a decade since I last used Solid Edge, so take my description of the steps with a grain of salt.